614 CHAPTER 9 Polar Coordinates; Vectors Now Work PROBLEM 21 Figure 28 r x y 4sin , or 2 4 2 2 θ ( ) = + − = x O y 2 4 5 u 5 0 u 5 p u = 1 3 1 2 3 4 5 p– 2 u 5 3p –– 2 u 5 7p –– 4 u 5 p– 4 u 5 3p –– 4 u 5 5p –– 4 (a) 20.5 (b) 24 4 4.5 Identifying and Graphing a Polar Equation (Circle) Identify and graph the equation: θ = − r 2 cos EXAMPLE 7 Identifying and Graphing a Polar Equation (Circle) Identify and graph the equation: r 4sinθ = Solution EXAMPLE 6 To transform the equation to rectangular coordinates, multiply each side by r. r r4 sin 2 θ = Now use the facts that r x y 2 2 2 = + and y rsin .θ = Then x y y x y y x y y x y 4 4 0 4 4 4 2 4 2 2 2 2 2 2 2 2 ( ) ( ) ( ) + = + − = + − + = + − = This is the standard equation of a circle with center at 0, 2 ( ) in rectangular coordinates and radius 2. Figure 28(a) shows the graph drawn by hand. Figure 28(b) shows the graph using a TI-84 Plus CE with min 0, max 2 , θ θ π = = and step 24 . θ π = Complete the square in y. Factor. THEOREM Let a be a nonzero real number. Then the graph of the equation θ = r a sin is a horizontal line. It lies a units above the pole if a 0 > and lies a units below the pole if a 0. < The graph of the equation θ = r a cos is a vertical line. It lies a units to the right of the pole if a 0 > and lies a units to the left of the pole if a 0. <
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